Because DTS is arm sampling under DTS is context independent (see (REF )), we can analyze its behavior as if contexts are observed only after an arm is sampled.
With this view, we have written down a model where, DTS selects and action \(I_t\)  and then observes the context \(X_t\)  and reward \(R_t=\langle X_t \,,\, \theta ^{(I_t)} \rangle + W_t\) . An alternative description of the probability space would generate all the randomness upfront, using what [1]} call a reward table and a context table. Precisely, let \(\tilde{X}_{n,i}\sim p_X\)  and \(\tilde{W}_{n,i}\sim N(0, \sigma ^2)\)  denote virtual contexts and rewards drawn i.i.d across \(n\in \mathbb {N}\)  and \(i\in [k]\) . Assume \((\tilde{X}_{n,i})_{n\in \mathbb {N},i\in [k]}\) , \((\tilde{R}_{n,i})_{n\in \mathbb {N},i\in [k]}\)  and the algorithm's internal randomness (used for arm sampling) are jointly independent. Let \(\tilde{R}_{n,i}= \langle \theta ^{(i)} \, , \, \tilde{X}_{n,i} \rangle + \tilde{W}_{n,i}\)  denote the reward corresponding to the \(n\) th play of arm \(i\) . Then for \(t\in \mathbb {N}\)
\((I_t, X_{t}, R_t) \overset{D}{=} \left(I_t, \tilde{X}_{N_{t, I_t}+1, I_t}, \tilde{R}_{N_{t, I_t}+1, I_t}\right)\) 
